Volume bounds for shadow covering

Chen, Christina; Khovanova, Tanya; Klain, Daniel

doi:10.1090/S0002-9947-2013-05855-2

Volume bounds for shadow covering
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by Christina Chen, Tanya Khovanova and Daniel A. Klain PDF

Trans. Amer. Math. Soc. 366 (2014), 1161-1177 Request permission

Abstract:

For $n \geq 2$ a construction is given for a large family of compact convex sets $K$ and $L$ in $\mathbb {R}^n$ such that the orthogonal projection $L_u$ onto the subspace $u^\perp$ contains a translate of the corresponding projection $K_u$ for every direction $u$, while the volumes of $K$ and $L$ satisfy $V_n(K) > V_n(L).$

It is subsequently shown that if the orthogonal projection $L_u$ onto the subspace $u^\perp$ contains a translate of $K_u$ for every direction $u$, then the set $\frac {n}{n-1} L$ contains a translate of $K$. It follows that \[ V_n(K) \leq \left (\frac {n}{n-1}\right )^n V_n(L).\] In particular, we derive a universal constant bound \[ V_n(K) \leq 2.942 V_n(L),\] independent of the dimension $n$ of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.

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